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Boolean ring

In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists of idempotent elements. These rings arise from (and give rise to) Boolean algebras. One example is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection.

Every Boolean ring R satisfies x + x = 0 for all x in R, because we know

x + x = (x + x)2 = x2 + 2x2 + x2 = x + 2x + x

and we can subtract x + x from both sides of this equation. A similar proof shows that every Boolean ring is commutative:

x + y = (x + y)2 = x2 + xy + yx + y2 = x + xy + yx + y

and this yields xy + yx = 0, which means xy = −yx = yx (using the first property above).

If we define

x &and y = xy,

xy = x + yxy,

~x = 1 + x

then these satisfy all of the axioms for meets, joins, and complements in a Boolean algebra. Thus every Boolean ring with 1 becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring with 1 thus:

xy = xy,

x + y = (xy) ∧ ~(xy).